1. BESC 320 (Water & Bioenvironmental Science) Intro to probabilistic thought Statistics answers questions: It gives measures of effect size It sets confidence limits on conclusions It is simple in principles and general practice Therefore, it is a crime to NOT be an expert E.g. two simple tests 1.A lady is asked in eight instances whether milk or tea is added first to her coffee 2.The lady is asked to rate how much she enjoys each cup on a scale of 1-100

2. Ronald Aylmer Fisher (1890-1962) Founding father of modern statistics and the Darwinian synthesis. Book rec: The Origins of Theoretical Population Genetics, by Will Provine ($10) In 1919 worked as a statistician at the Rothamsted Agricultural Experiment Station in the UK. Published many papers and wrote several books on experimental design and evolution. Creative demonstration of powers of statistical analysis using data from a “Lady Tasting Tea”.

3. 1.A lady is asked in eight instances whether milk or tea is added first Correct Incorrect Actual data 8 0 Random expectation 4 4 Calculate degree of pattern (deviation from random) and if improbably large then bias (causation) exists. That is, you conclude she can tell the difference. If pattern could reasonably be due to chance alone then accept the default (null) hypothesis that she can not tell (i.e. unpatterned data) 2.The lady rates how much she enjoys each cup on a scale of 1-100 Calculate deviation from random (using trick of comparing between- and within-group variance. Both approaches contrast Yin of Pattern with Yang of Random ☯

4. At Rothamsted Fisher recognized problems with agricultural experiments Fisher’s Solution: Replicate, Randomize (Spread variation without bias among treatments) Source of Picture: http://www.ipm.iastate.edu/ipm/icm/files/images/uneven-corn-VS6.jpg Same field, same treatment, but plant performance is uneven... Thick Growth Thin Growth

5. Fisher’s Lessons from Rothamsted Experiments prior to Fisher generally involved two fields (containing hundreds of plants), each receiving a treatment (e.g. two levels of N) Problem: So much variability exists within each field it is difficult or impossible to tease out the treatment effect (i.e. a signal to noise problem) Field with High N Field with Low N Growth Treatment

6. Fisher’s Solution at Rothamsted – Old Problematic Design: One large field receiving high nitrogen (N), one large field receiving low nitrogen (N). (Today this design is sometimes called “pseudoreplication” if the experimenter attempts to say that the sample size is the number of plants.) – New Improved Design: Many small plots, randomly receiving high N or low N; plots can also be blocked to help tease out the variation due to location and local conditions. Hurlbert, S. H. (1984). Pseudoreplication and the design of ecological field experiments. Ecological monographs 54(2): 187-211.

7. Examples of Correct & Incorrect Ways to Randomize Treatments Correct Ways: Use a random number table. Pick treatments from a hat. Flip a coin. Incorrect Ways: Haphazardly decide which experimental units should receive which treatments. (Problem: too tempting for experimenter to bias.) Use a net to grab the goldfish in an ecology study. (Problem: might pick just the easiest to catch, sickly animals.) Alternate treatments (every other one). (Problem: that’s systematic, not random; who knows what other factors vary in the same systematic way.) Assign people to drug study on the basis of their last name. (Problem: could be related to a person’s ancestry.)

8. Fisher, Randomization, Replication & Blocking No replication (or pseudoreplication) (Rothamsted, pre-Fisher): Replicated with complete randomization: Replicated, randomized and blocked design: Field with High N Field with Low N Field broken up into smaller plots Plots are blocked by location or other condition; treatments are applied randomly to plots within blocks. Field broken up into smaller plots & plots are grouped. Treatments are applied to plots rather than to an entire field; this improves replication & interspersion of treatments. Dashed rectangle is a block

9. Another of Fisher’s Contributions to Statistics: The Analysis of Variance (ANOVA) Allows scientists to mathematically partition variation in a measured variable due to different sources (treatments, blocks, plots, for example). Some of Fisher’s contributions to the field of statistics grew out of his experience with spatial agricultural experiments at Rothamsted.

10. At Rothamsted, Fisher saw firsthand that the purpose of good experimental design is not to eliminate variation entirely, but rather to try to ensure that extraneous variation is spread evenly among treatments. In the case of ANOVA, the experimental design can enable the variation to be partitioned mathematically during analysis. Variation in growth of plants can be partitioned into different sources of variation: 1. Variation in soil moisture, texture, etc. within a plot. 2. Variation between treatments (high N and low N). 3. Variation in soil moisture, texture, sunlight, etc., among blocks. Why do these two plants differ in growth? Is it because of block, treatment, or extraneous variation within plots?

11. This and following slides by TJ DeWitt Let us try an experiment and analysis Fiji water is awesome Everyone knows that Let’s prove it Fiji water is awesome Everyone knows that Let’s prove it Two tests: I.Side by side comparisonII. Scaled measure of quality

12. 1.Chi-square (χ²) on our water preference test data Fiji RO (remineralized) Actual data 22 31 Random expectation 26.5 26.5 (expectation of 53 random outcomes) Calculate deviation (bias) from random and if improbably large students have a patterned taste preference, else do not make that conclusion. χ² = ∑ (obs-exp)² = (22-26.5)²/26.5 + (31-26.5)²/26.5 = 1.528 exp The probability, P, of getting a metric of pattern this great due only to chance is 0.216—not improbable. Generally if P < 0.05 we consider the pattern Improbable due to chance. Thus we are safest concluding there is insufficient evidence of pattern here; i.e., no taste preference noted. FYI: Get P values in Excel® for χ² tests by entering, e.g., “=CHIDIST(1.528,1)” into a cell.

13. 2. t-test on our water preference data Data at left (you can paste into Excel®). Random expectation: average difference of 0 between water taste scores given by students for Fiji and RO Recall measures of pattern in statistics pit the among group deviations scaled to within group deviations. Here our measure of pattern is a t statistic—the average difference between scores divided by the standard deviation of within-individual differences: t = avg1-avg2 = 66.94 - 77.11 = 0.04 stdev(diffs)/sqrt(n) 30.95 / 6.86 Not big. The P-value is 0.97. It would be common to get a measure of pattern this large (or larger) by chance. 6485 5080 4080 9585 7065 5550 10075 5080 7060 50 8090 7142 0100 7580 1005 5070 8570 9095 7570 7550 7085 8070 056 8560 8060 10075 8565 6090 6735 7050 8020 10 6040 2163 7570 6590 1060 5060 7585 100 8073 8867 6080 100 50 9080 9078.5

14. So what are the cardinal points? The field of statistics provides tools to measure pattern against random (or a priori) expectations Test statistics, like χ², t, F, Λ, are metrics of pattern 1. generally among group (or along gradient) variation relative to within group (or off gradient) variation 2. Can be compared to the greatest expected values of the test statistics one might expect to arise by chance alone 3. A P value is the chance of a pattern equal to or greater than that observed occurring only by chance Independent replication is important in statistical analysis so pattern due to sloppy experimental design can not intrude to create either excess bias or noise.